Certainly! Let's solve the problem step-by-step:
We know the total perimeter of Stanley's fence is 44 feet. The formula for the perimeter [tex]\( P \)[/tex] of a rectangle is:
[tex]\[ P = 2L + 2W \][/tex]
where [tex]\( L \)[/tex] is the length and [tex]\( W \)[/tex] is the width of the rectangle.
We are also told that the width is two feet less than half the length. This can be expressed as:
[tex]\[ W = \frac{1}{2}L - 2 \][/tex]
Now, we can substitute this expression for [tex]\( W \)[/tex] into the perimeter equation:
[tex]\[ 2L + 2\left(\frac{1}{2}L - 2\right) = 44 \][/tex]
Let's simplify this step-by-step:
1. Distribute the 2 across the terms in the parentheses:
[tex]\[ 2L + 2 \cdot \frac{1}{2}L - 2 \cdot 2 = 44 \][/tex]
2. This simplifies to:
[tex]\[ 2L + L - 4 = 44 \][/tex]
3. Combine the like terms:
[tex]\[ 3L - 4 = 44 \][/tex]
4. Add 4 to both sides to isolate the term with [tex]\( L \)[/tex]:
[tex]\[ 3L = 48 \][/tex]
5. Divide both sides by 3 to solve for [tex]\( L \)[/tex]:
[tex]\[ L = \frac{48}{3} \][/tex]
[tex]\[ L = 16 \][/tex]
So, the length of the fence is 16 feet.
Now, we can find the width. Recall the expression for [tex]\( W \)[/tex]:
[tex]\[ W = \frac{1}{2}L - 2 \][/tex]
Substitute [tex]\( L = 16 \)[/tex] into the expression:
[tex]\[ W = \frac{1}{2} \cdot 16 - 2 \][/tex]
[tex]\[ W = 8 - 2 \][/tex]
[tex]\[ W = 6 \][/tex]
Therefore, the width of the fence is 6 feet.
